Am I killing my iPhone battery?

I recently got an iPhone 5s. I'm doing the same with it as I did my old phone - leaving it on charge the whole night so it's full battery in the morning. Now obviously the 5s has that super fast charger, so it will be 100% in like 2 hours and left like that while still plugged in for the rest of the night. Will this kill the battery life?

Yes and no. When your phone is plugged in like that, it is constantly going down to 99% and up to 100%. After it does this 100 times it counts as one cycle. Assuming you do this on a normal charger for 6 hours and you get 1% discharge every 5-10 min, then maybe a 1-2 min to recharge that is about 36%, so about 1/3 full phone charges. Your phone is rated at about 1000 discharges before capacity goes down to 80% of the factory capacity. Assuming you are on a 2 year contract, that is about 262 cycles, plus the normal 1 cycle a day is about 1000 cycles so battery will be about 80% of where it was when you get a new phone.